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A145356 Partition number array, called M31hat(6).

%I A145356 #18 May 27 2025 10:31:02
%S A145356 1,6,1,42,6,1,336,42,36,6,1,3024,336,252,42,36,6,1,30240,3024,2016,
%T A145356 1764,336,252,216,42,36,6,1,332640,30240,18144,14112,3024,2016,1764,
%U A145356 1512,336,252,216,42,36,6,1,3991680,332640,181440,127008,112896,30240,18144,14112
%N A145356 Partition number array, called M31hat(6).
%C A145356 Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;n,k) with the k-th partition of n in A-St order.
%C A145356 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A145356 Sixth member (K=6) in the family M31hat(K) of partition number arrays.
%C A145356 If M31hat(6;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(6):= A145357.
%H A145356 Wolfdieter Lang, <a href="/A145356/a145356.txt">First 10 rows of the array and more.</a>
%H A145356 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial Interpretation of Generalized Stirling Numbers</a>, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
%F A145356 a(n,k) = product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)| = A049374(n,1) = A001725(n+4) = [1,6,42,336,3024,30240,332640,...] = (n+4)!/5!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.
%e A145356 Triangle begins
%e A145356   [1];
%e A145356   [6,1];
%e A145356   [42,6,1];
%e A145356   [336,42,36,6,1];
%e A145356   [3024,336,252,42,36,6,1];
%e A145356   ...
%e A145356 a(4,3)= 36 = |S1(6;2,1)|^2. The relevant partition of 4 is (2^2).
%Y A145356 Cf. A145358 (row sums).
%Y A145356 Cf. A144890 (M31hat(5) array), A145357 (S1hat(6)).
%K A145356 nonn,easy,tabf
%O A145356 1,2
%A A145356 _Wolfdieter Lang_, Oct 17 2008, Oct 28 2008